Sure, let's solve the system of equations step-by-step:
Given the system of equations:
1. [tex]\( y = x^2 - 5x - 4 \)[/tex]
2. [tex]\( y = -2x \)[/tex]
To find the ordered pair solutions, we can substitute the expression for [tex]\( y \)[/tex] from the second equation into the first equation.
So, substitute [tex]\( y = -2x \)[/tex] into [tex]\( y = x^2 - 5x - 4 \)[/tex]:
[tex]\[ -2x = x^2 - 5x - 4 \][/tex]
Now, let's rearrange this equation to form a standard quadratic equation:
[tex]\[ x^2 - 5x - 4 + 2x = 0 \][/tex]
[tex]\[ x^2 - 3x - 4 = 0 \][/tex]
Next, we'll solve the quadratic equation [tex]\( x^2 - 3x - 4 = 0 \)[/tex]. We can do this by factoring:
[tex]\[ x^2 - 3x - 4 = (x - 4)(x + 1) = 0 \][/tex]
Setting each factor to zero gives us:
[tex]\[ x - 4 = 0 \quad \text{or} \quad x + 1 = 0 \][/tex]
[tex]\[ x = 4 \quad \text{or} \quad x = -1 \][/tex]
Now, we need to find the corresponding [tex]\( y \)[/tex] values for each [tex]\( x \)[/tex].
For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -2(4) = -8 \][/tex]
So, one solution is [tex]\( (4, -8) \)[/tex].
For [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -2(-1) = 2 \][/tex]
So, the other solution is [tex]\( (-1, 2) \)[/tex].
Therefore, the ordered pair solutions for the system of equations are:
[tex]\[ (-1, 2) \][/tex] and [tex]\[ (4, -8) \][/tex].
